Theories of Information Processing
From quantum foundations, via meaning in
natural language, to a theory of everything
Bob Coecke
University of Oxford
Our starting point is the common structure of:
Our starting point is the common structure of:
• how quantum systems interact
• how meanings in natural language interact
Initial papers:
• S. Abramsky & BC (2004) A categorical semantics of quantum protocols.
LiCS’04. arXiv:quant-ph/0402130
• BC, M. Sadrzadeh & S. Clark (2010) Mathematical foundations for a
compositional distributional model of meaning. arXiv:1003.4394
Focus on common structure:
• BC (2012) The logic of quantum mechanics - Take II. arXiv:1204.3458
• S. Clark, BC, E. Grefenstette, S. Pulman & M. Sadrzadeh (2013) A quantum teleportation inspired algorithm produces sentence meaning from word
meaning and grammatical structure. arXiv:1305.0556
– e.g. entanglement swapping –
– e.g. negation and relative pronouns –
Claim. Also in the scope of the framework are:
• human behaviour w.r.t. institutions,
• animal behaviour w.r.t. evolution, etc.
Forthcoming paper:
• BC (2015) From quantum foundations, via meaning in natural language, to a theory of everything. In: The Incomputable, S. B. Cooper & S. Soskova, Eds. Springer.
Initial question (quant-ph/0510032):
Initial question (quant-ph/0510032):
Can QM be formulated in pictures?
Initial question (quant-ph/0510032):
Can QM be formulated in pictures?
Same question, put differently:
• Can QM be formulated in terms of composition?
(contra: C, +, matrices, ...)
Initial question (quant-ph/0510032):
Can QM be formulated in pictures?
Same question, put differently:
• Can QM be formulated in terms of composition?
(contra: C, +, matrices, ...)
• Can QM be formulated in terms of processes?
(contra: states, numbers)
Initial question (quant-ph/0510032):
Can QM be formulated in pictures?
Same question, put differently:
• Can QM be formulated in terms of composition?
(contra: C, +, matrices, ...)
• Can QM be formulated in terms of processes?
(contra: states, numbers)
• Does QM have logic features?
(contra: failures)
BC & Aleks Kissinger
Picturing Quantum Processes
Cambridge University Press, fall 2015, 800 pp.
— Ch. 1 – Processes as diagrams —
Philosophy [i.e. physics] is written in this grand book—I mean the
universe—which stands continually open to our gaze, but it cannot be
understood unless one first learns to comprehend the language and
interpret the characters in which it is written. It is written in the
language of mathematics, and its characters are triangles, circles,
and other geometrical figures, without which it is humanly impossible
to understand a single word of it; without these, one is wandering
around in a dark labyrinth.
— Galileo Galilei, “Il Saggiatore”, 1623.
Here we introduce:
• diagrams
• process theories
— Ch. 1 – Processes as diagrams —
– processes as boxes and systems as wires –
— Ch. 1 – Processes as diagrams —
– processes as boxes and systems as wires –
— Ch. 1 – Processes as diagrams —
– processes as boxes and systems as wires –
— Ch. 1 – Processes as diagrams —
– processes as boxes and systems as wires –
— Ch. 1 – Processes as diagrams —
– processes as boxes and systems as wires –
— Ch. 1 – Processes as diagrams —
– composing processes –
— Ch. 1 – Processes as diagrams —
– composing processes –
— Ch. 1 – Processes as diagrams —
– composing processes –
— Ch. 1 – Processes as diagrams —
– process theory –
... consists of:
• set of systems S
• set of processes P
which are:
• closed under forming diagrams.
— Ch. 1 – Processes as diagrams —
– process theory –
... consists of:
• set of systems S
• set of processes P
which are:
• closed under forming diagrams.
It tells us:
• how to interpret boxes and wires,
• and hence, when two diagrams are equal.
— Ch. 1 – Processes as diagrams —
– process theory –
— Ch. 1 – Processes as diagrams —
– process theory –
— Ch. 1 – Processes as diagrams —
– special processes/diagrams –
— Ch. 1 – Processes as diagrams —
– special processes/diagrams –
State :=
— Ch. 1 – Processes as diagrams —
– special processes/diagrams –
State :=
Effect / Test :=
— Ch. 1 – Processes as diagrams —
– special processes/diagrams –
State :=
Effect / Test :=
Number :=
— Ch. 1 – Processes as diagrams —
– special processes/diagrams –
Dirac notation :=
— Ch. 1 – Processes as diagrams —
– special processes/diagrams –
Born rule :=
... what about natural language meaning?
Vector space model of word meaning:
• core of most natural language processing (NLP)
• requires a huge corpus of text (e.g. internet)
Vectors as word meanings:
• vector space spanned by context words
• meaning vectors from relative occurrences
• similarity from ‘Born rule’
H. Schuetze (1998) Automatic word sense discrimination. Computational Linguistics, 24, 97123.
Vector space model of word meaning:
• core of most natural language processing (NLP)
• requires a huge corpus of text (e.g. internet)
Vectors as word meanings:
• vector space spanned by context words
• meaning vectors from relative occurrences
• similarity from ‘Born rule’
H. Schuetze (1998) Automatic word sense discrimination. Computational Linguistics, 24, 97123.
... what about sentence meaning?
... where is the grammar?
— Ch. 2 – String diagrams —
When two systems, of which we know the states by their respective
representatives, enter into temporary physical interaction due to known
forces between them, and when after a time of mutual influence the
systems separate again, then they can no longer be described in the
same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic
trait of quantum mechanics, the one that enforces its entire departure
from classical lines of thought.
— Erwin Schrödinger, 1935.
Here we introduce:
• string diagrams
• transposes and adjoints
• quantum phenomena in great generality
— Ch. 2 – String diagrams —
– TFAE –
— Ch. 2 – String diagrams —
– TFAE –
1. Causal diagrams with cup-state and cap-effect:
which satisfy:
— Ch. 2 – String diagrams —
– TFAE –
2. diagrams allowing in-in, out-out and out-in wiring:
— Ch. 2 – String diagrams —
– TFAE –
From 1. to 2.:
— Ch. 2 – String diagrams —
– TFAE –
From 1. to 2.:
so that:
— Ch. 2 – String diagrams —
– transpose –
— Ch. 2 – String diagrams —
– transpose –
... :=
— Ch. 2 – String diagrams —
– transpose –
Clever new notation:
— Ch. 2 – String diagrams —
– transpose –
Clever new notation:
(intuition: again yanking the wire)
— Ch. 2 – String diagrams —
– transpose –
Prop. Sliding:
— Ch. 2 – String diagrams —
– transpose –
Prop. Sliding:
... so this is a mathematical equation:
— Ch. 2 – String diagrams —
– adjoints –
From a state to its test:
— Ch. 2 – String diagrams —
– adjoints –
— Ch. 2 – String diagrams —
– adjoint & conjugate –
Unitarity/isometry :=
— Ch. 2 – String diagrams —
– quantum teleportation –
— Ch. 2 – String diagrams —
– quantum teleportation –
— Ch. 2 – String diagrams —
– quantum teleportation –
— Ch. 2 – String diagrams —
– quantum teleportation –
... what about natural language meaning?
String diagrams for natural language meaning:
String diagrams for natural language meaning:
• Top part: grammar
• Bottom part: meaning vectors
Mathematics of grammar:
Lambek’s Residuated monoids (1950’s):
b≤a(c⇔a·b≤c⇔a≤cb
or equivalently,
a · (a ( c) ≤ c ≤ a ( (a · c)
(c b) · b ≤ c ≤ (c · b) b
Lambek’s Pregroups (2000’s):
a · −1a ≤ 1 ≤ −1a · a
b−1 · b ≤ 1 ≤ b · b−1
From grammar to meaning:
For noun type n, verb type is −1n · s · n−1, so:
From grammar to meaning:
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
From grammar to meaning:
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
From grammar to meaning:
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
From grammar to meaning:
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
Diagrammatic type reduction:
From grammar to meaning:
For noun type n, verb type is −1n · s · n−1, so:
n · −1n · s · n−1 · n ≤ 1 · s · 1 ≤ s
Diagrammatic type reduction:
Algorithm for meaning composition:
1. Perform grammatical type reduction:
(word type 1) . . . (word type n) { sentence type
2. Interpret diagrammatic type reduction as linear map:




X
X







7→  hii| ⊗ id ⊗  hii|
f ::
i
i
3. Apply this map to tensor of word meaning vectors:
→
−
→
−
f v ⊗ ... ⊗ v
1
n
Algorithm for meaning composition:
Dimitri Kartsaklis & Mehrnoosh Sadrzadeh (2013) Prior Disambiguation of
Word Tensors for Constructing Sentence Vectors. In EMNLP’13.
Algorithm for meaning composition:
1. Perform grammatical type reduction:
(word type 1) . . . (word type n) { sentence type
2. Interpret diagrammatic type reduction as linear map:




X
X







7→  hii| ⊗ id ⊗  hii|
f ::
i
i
3. Apply this map to tensor of word meaning vectors:
→
−
→
−
f v ⊗ ... ⊗ v
1
n
Algorithm for social behaviour composition:
1. Perform social type reduction:
(person type 1) . . . (person type n) { group type
2. Interpret diagrammatic type reduction as linear map:




X
X







7→  hii| ⊗ id ⊗  hii|
f ::
i
i
3. Apply this map to tensor of word meaning vectors:
→
−
→
−
f v ⊗ ... ⊗ v
1
n
speaking about ≡ reasoning about
... what about ambiguity?
— Ch. 4 – Quantum processes —
The art of progress is to preserve order amid change, and to preserve
change amid order.
— Alfred North Whitehead, Process and Reality, 1929.
Here we introduce:
• quantum maps
• mixing
— Ch. 4 – Quantum processes —
– pure quantum maps –
Born-rule :=
— Ch. 4 – Quantum processes —
– pure quantum maps –
... :=
— Ch. 4 – Quantum processes —
– quantum maps –
Discarding :=
— Ch. 4 – Quantum processes —
– quantum maps –
... := pure quantum maps + discarding
Mixing:
Mixing:
Two distinct sums:
Density matrix model of meaning:
• Ambiguity as mixedness:
Density matrix model of meaning:
• Ambiguity as mixedness:
• Disambiguation :=
Density matrix model of meaning:
• Ambiguity as mixedness:
• Disambiguation :=
Density matrix model of meaning:
• Ambiguity as mixedness:
• Disambiguation :=
Density matrix model of meaning:
• Ambiguity as mixedness:
• Disambiguation :=
Density matrix model of meaning:
• Ambiguity as mixedness:
• Disambiguation :=
... what are those dots?
— Ch. 6 – Picturing classical processes —
Damn it! I knew she was a monster! John! Amy! Listen! Guard your
buttholes.
— David Wong, This Book Is Full of Spiders, 2012.
Here we fully diagrammatically describe:
• classical data as spiders
• fully diagrammatic protocols
— Ch. 6 – Picturing classical processes —
– classical-quantum maps –
Main idea:
single wire
classical system
=
quantum system double wire
— Ch. 6 – Picturing classical processes —
– spiders –
... :=
— Ch. 6 – Picturing classical processes —
– teleportation diagrammatically –
— Ch. 6 – Picturing classical processes —
– teleportation diagrammatically –
— Ch. 6 – Picturing classical processes —
– teleportation diagrammatically –
Relative pronouns:
M. Sadrzadeh, BC & S. Clark (2013 & 2014) The Frobenius anatomy of word
meaning I & II. Journal of Logic and Computation.
ρ she



|AliceihAlice|


P

|BethihBeth|
:= 




 ...
ρhates

0


|AliceihAlice|
⊗
ρ
⊗ |BobihBob|



P
|BethihBeth| ⊗ ρ00 ⊗ |ColinihColin|
:= 




 ...
ρBob := |BobihBob|
ρ she



|AliceihAlice|


P

|BethihBeth|
:= 




 ...
ρhates

0


|AliceihAlice|
⊗
ρ
⊗ |BobihBob|



P
|BethihBeth| ⊗ ρ00 ⊗ |ColinihColin|
:= 




 ...
ρBob := |BobihBob|
ρ sentence := |AliceihAlice|
speaking about ≡ reasoning about